High-accuracy angular encoders are widely used for rotational axis position measurement in ultra-precision manufacturing equipment and measurement instruments, such as diamond turning machines for free-form optical surface generation. A typical rotary encoder is made of a glass disk with a fine grating pattern and a scanning read head unit. Digital pulse trains are output from the read head to represent the disk rotary position. The angular measurement accuracy of such a rotary encoder is highly dependent on the grating pattern manufacturing error (uniformity and eccentricity of the graduation), encoder disk installation eccentricity on the rotary axis, read head alignment, and so on. Repeatable components of encoder error can in principle be eliminated through calibration to improve accuracy. Such calibration methods are classified into two categories: comparison calibration and self-calibration.
Comparison calibration is based on measuring the difference between a higher-accuracy angle standard and the axis under calibration. Polygon mirrors are often employed as angle standards, and the difference is usually measured by photo-electric autocollimators. However, the number of calibration points is limited by the polygon facet number. Renishaw has commercialized a rotary axis calibration system that is composed of a 72-tooth index table with 1 arc-sec accuracy and an angle-measurement laser interferometer (Renishaw User Manual, 2002, Rotary Axis Calibration Using the RX10 Rotary Indexer). As this system requires the index table to unlock and lock down at every tooth position, it is time-consuming and the accuracy is limited by both the index table and the interferometer. As an alternative approach, a ring-laser based goniometer can be used to calibrate rotary axes (Filatov, Y. V., Loukianov, D. P., Probst, R., 1997, Dynamic Angle Measurement by Means of a Ring Laser, Metrologia, 34:343-351).
To avoid the use of external standards and associated errors, many self-calibration methods have been developed. Circle closure has been frequently used to cross-calibrate index tables at Moore Special Tool (Moore, W. R., 1970, Foundations of Mechanical Accuracy, The Moore Special Tool Co., Bridgeport, Conn., USA) and at NIST (Estler, W. T., Queen, Y. T., 1993, An Advanced Angle Metrology System, Annals of the CIRP, 42/1:573-576 and Estler, W. T., 1998, Uncertainty Analysis for Angle Calibrations Using Circle Closure, J. Res. Natl. Inst. Stand. Technol., 103/2: 141-151). The calibrated tables can be used as a reference standard in a comparison calibration system. Masuda and Kajitani (Masuda, T., Kajitani, M., 1989, An Automatic Calibration System for Angular Encoders, Precision Engineering, 11/2:95-100) developed a calibration system for angular encoders in which six read-heads were unevenly spaced around a master scale to self-check its accuracy. This self-calibrated master scale was then used to calibrate other encoders installed on the system axis via comparison. Also, Watanabe et al (Watanabe, T., Fujimoto, H., Nakayama, K., Masuda, T., Kajitani, M., 2003, Automatic High Precision Calibration System for Angular Encoder, Proc. of SPIE, 5190:400-409) further developed an angle encoder calibration system that automatically derives Fourier components of encoder error by cross-checking graduation errors of two coaxially installed encoder disks at multiple read-head positions. Based on this principle, a simplified system was presented using five read-heads evenly distributed around a single optical disk such that graduation errors excepting 5th order components were self-calibrated (Watanabe, T., Fujimoto, H., Masuda, T., 2005, Self-Calibration Rotary Encoder, 7th Int., Sym, Meas., Techol. Intellig. Instrum., 240-245). Zhang and Li (Zhang, G. X., Wang, C. H., Li, Z., 1994, Improving the Accuracy of Angle Measurement System with Optical Grating, Annals of the CIRP, 43/1:457-460) presented an angle measurement system with sine function transmissivity grating disks and four unevenly spaced read-heads. At PTB (Probst, R., Wittekopf, R., Krause, M., Dangschat, H., Ernst, A., 1998, The New PTB Angle Comparator, Meas. Sci. Technol., 9:1059-1066), an angle comparator was constructed using 15 read-heads to achieve 0.01 arc-sec uncertainty. In this system, the encoder errors at 128 circle divisions can be self-calibrated. Using Fourier series, Geckeler (Geckeler, R. D., Fricke, A., Elster, C., 2006, Calibration of Angle Encoders Using Transfer Functions, Meas. Sci. Technol., 17:2811-2818) described a calibration method using two rotary tables and a polygon mirror. In all the above references, self-calibration methods were developed for special encoder and spindle setups (typically using multiple read-heads), but these methods cannot be directly applied to calibrate general encoders with a single read-head. Further, the encoders under calibration are usually required to be installed on the calibration system axis.
Ideally, encoders should be ideally calibrated on their own application axes because significant errors are introduced by the disk installation and read-head alignment. In Michaelis, T. D., 1986, Apparatus and Method for Generating Calibrated Optical Encoder Pulses, U.S. Pat. No. 4,593,193, a method based on an air-bearing spindle in free rotation was proposed to correct encoder pulses. However, the calibration result can only be applied to the speed used in calibration. In Orton, P. A., Poliakoff, J. F., Hatiris, E., Thomas, P. D., 2001, Automatic Self-Calibration of an Incremental Motion Encoder, 2001 IEEE Instrum. Meas. Technol. Conf., 1617-1618, a self-calibration method was described to measure disk graduation error at constant spindle speed. However, it is impractical to control the spindle speed accurately enough for calibration purposes, due to imbalance, feedback sensor errors, and motor driving torque ripple.